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Estimating the Marginal Law of a Time Series with Applications to Heavy Tailed Distributions

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  • Christian Francq

    () (CREST)

  • Jean-Michel Zakoïan

    () (CREST)

Abstract

This article addresses estimating parametric marginal densities of stationary time series in the absence of precise information on the dynamics of the underlying process. We propose using an estimator obtained by maximization of the "quasi-marginal" likelihood, which is a likelihood written as if the observations were independent. We study the effect of the (neglected) dynamics on the asymptotic behavior of this estimator. The consistency and asymptotic normality of the estimator are established under mild assumptions on the dependence structure. Applications of the asymptotic results to the estimation of stable, generalized extreme value and generalized Pareto distributions are proposed. The theoretical results are illustrated on financial index returns. Supplementary materials for this article are available online.
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Suggested Citation

  • Christian Francq & Jean-Michel Zakoïan, 2011. "Estimating the Marginal Law of a Time Series with Applications to Heavy Tailed Distributions," Working Papers 2011-30, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2011-30
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    References listed on IDEAS

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    1. Jansen, Dennis W & de Vries, Casper G, 1991. "On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective," The Review of Economics and Statistics, MIT Press, vol. 73(1), pages 18-24, February.
    2. Gamini Premaratne, 2005. "A Test for Symmetry with Leptokurtic Financial Data," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 3(2), pages 169-187.
    3. Boubacar Mainassara, Y. & Carbon, M. & Francq, C., 2012. "Computing and estimating information matrices of weak ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 345-361.
    4. Cotter, John, 2007. "Varying the VaR for unconditional and conditional environments," Journal of International Money and Finance, Elsevier, vol. 26(8), pages 1338-1354, December.
    5. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    6. Einmahl, John H. J. & Li, Jun & Liu, Regina Y., 2009. "Thresholding Events of Extreme in Simultaneous Monitoring of Multiple Risks," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 982-992.
    7. Stephen J. Taylor, 2007. "Introduction to Asset Price Dynamics, Volatility, and Prediction," Introductory Chapters,in: Asset Price Dynamics, Volatility, and Prediction Princeton University Press.
    8. Shiqing Ling & Michael McAleer, 2010. "A general asymptotic theory for time-series models," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 97-111.
    9. Francq, Christian & Roy, Roch & Zakoian, Jean-Michel, 2005. "Diagnostic Checking in ARMA Models With Uncorrelated Errors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 532-544, June.
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    Cited by:

    1. repec:spr:metrik:v:81:y:2018:i:6:d:10.1007_s00184-018-0674-z is not listed on IDEAS
    2. Auray, Stéphane & Eyquem, Aurélien & Jouneau-Sion, Frédéric, 2014. "Modeling tails of aggregate economic processes in a stochastic growth model," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 76-94.
    3. Fries, Sébastien & Zakoian, Jean-Michel, 2017. "Mixed Causal-Noncausal AR Processes and the Modelling of Explosive Bubbles," MPRA Paper 81345, University Library of Munich, Germany.
    4. Delaigle, Aurore & Meister, Alexander & Rombouts, Jeroen, 2016. "Root-T consistent density estimation in GARCH models," Journal of Econometrics, Elsevier, vol. 192(1), pages 55-63.
    5. Yang, Yaxing & Ling, Shiqing, 2017. "Self-weighted LAD-based inference for heavy-tailed threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 197(2), pages 368-381.
    6. Preminger, Arie & Storti, Giuseppe, 2014. "Least squares estimation for GARCH (1,1) model with heavy tailed errors," MPRA Paper 59082, University Library of Munich, Germany.
    7. Echaust Krzysztof, 2014. "A Comparison of Tail Behaviour of Stock Market Returns," Folia Oeconomica Stetinensia, Sciendo, vol. 14(1), pages 1-13, June.

    More about this item

    Keywords

    alpha-stable distribution; composite likelihood; GEV distribution; GPD; pseudo-likelihood; quasi-marginal maximum likelihood; stock returns distributions;

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